Derivative of Inverse Csc

We now understand the derivative of inverse csc x. It is commonly denoted as d/dx (csc⁻¹ x) or (csc⁻¹ x)', and its value is -1/(|x|√(x²-1)). The function inverse csc x has a well-defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below: Inverse Cosecant Function: (csc⁻¹(x) is the inverse of csc(x)).

Chain Rule: A fundamental rule used in differentiating composite functions.

Absolute Value Function: |x| represents the absolute value of x.

The derivative of inverse csc x can be represented as d/dx (csc⁻¹ x) or (csc⁻¹ x)'.

The formula we use to differentiate inverse csc x is: d/dx (csc⁻¹ x) = -1/(|x|√(x²-1))

The formula applies to all x where |x| > 1.

We can derive the derivative of inverse csc x using proofs. To show this, we will use trigonometric identities along with differentiation rules. There are several methods we use to prove this, such as:

1. By Implicit Differentiation
2. Using Chain Rule

We will now demonstrate that the differentiation of inverse csc x results in -1/(|x|√(x²-1)) using the above-mentioned methods:

By Implicit Differentiation

To find the derivative of csc⁻¹ x using implicit differentiation, let y = csc⁻¹ x, which implies csc(y) = x. Differentiating both sides with respect to x, we get: d/dx (csc(y)) = d/dx (x) -csc(y)cot(y) dy/dx = 1

Solving for dy/dx: dy/dx = -1/(csc(y)cot(y))

Using the identity csc²(y) = 1 + cot²(y),

we have: dy/dx = -1/(x√(x²-1)) Since csc(y) = x,

we write the result as: dy/dx = -1/(|x|√(x²-1))

Hence, proved.

Using Chain Rule

To prove the differentiation of csc⁻¹ x using the chain rule, consider y = csc⁻¹ x.

Differentiate both sides: dy/dx = -1/(|x|√(x²-1))

This is consistent with the chain rule application, where the derivative of the inner function csc⁻¹ x is -1/(|x|√(x²-1)).

When a function is differentiated multiple times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be complex.

To understand them better, think of a scenario where the rate of change (first derivative) and the rate at which this rate changes (second derivative) are also changing. Higher-order derivatives enhance the understanding of functions like inverse csc(x).

For the first derivative of a function, we write f′(x), indicating how the function changes or its slope at a particular point. The second derivative is derived from the first derivative, denoted as f′′(x). Similarly, the third derivative, f′′′(x), is derived from the second derivative, and this pattern continues.

For the nth derivative of csc⁻¹(x), we generally use fⁿ(x) for the nth derivative of a function f(x) which tells us the change in the rate of change, continuing for higher-order derivatives.

When x = 1 or x = -1, the derivative is undefined because the function inverse csc(x) has vertical asymptotes at these points.

When x = √2, the derivative of csc⁻¹ x = -1/(|√2|√(2-1)), which simplifies to -1.

• Derivative: The derivative of a function quantifies how the function's output changes in response to a slight change in input.

• Inverse Cosecant Function: The inverse of the cosecant function, denoted as csc⁻¹(x), and defined for |x| > 1.

• Chain Rule: A rule in calculus for differentiating composite functions, crucial for finding the derivative of complex expressions.

• Absolute Value: The absolute value of a number, denoted as |x|, represents its non-negative value.

• Implicit Differentiation: A technique used to find the derivative of functions not solved for one variable in terms of another.